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When deriving DFT from DTFT,we sample the frequency domain with uniformly spaced samples,hence adding periodicity to time domain.

But DFT requires a limited support: we take only 1 period.

Does that make circular convolution the same as a linear convolution with the signal's periodic extension?

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Convolution in DFT is still circular.

Think of the DFT as taking the 1st period (in time and in frequency) of the DFS (discrete Fourier series). In DFS, both the time sequence and the frequency sequence are N-periodic, and the circular convolution applies beautifully.

I personally think all properties in terms of DFS, and then consider the 1st period when speaking DFT.

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    $\begingroup$ the DFT and the DFS are exactly the same thing. dsp.stackexchange.com/questions/16586/… dsp.stackexchange.com/questions/18144/… $\endgroup$ – robert bristow-johnson Sep 23 '17 at 17:52
  • $\begingroup$ @juancho: could you explain why circular convolution applies beautifully. I'm from statistics so linear convolution ( when computing a result of a filter and a signal ) makes sense to me. So, does correlation (although I'm not sure why DSPers connect convolution and correlation ). But circular convolution is still foggy for me. What is it doing is my basic question ? Thanks. $\endgroup$ – mark leeds Feb 3 '18 at 6:54

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